In 1936, Koebe proved that every planar graph is the incidence graph of a disk packing in . In higher dimensions, obstructions are to be expected. For instance,

Theorem 1 (Benjamini-Schramm) If is the incidence graph of a sphere packing in , then either is -parabolic, or has a non constant -harmonic function.

This motivates us to find properties which imply vanishing of reduced cohomology.

2. Definitions

Let be a graph. Denote by the set of functions with gradient in . Then reduced cohomology is

The Poisson boundary of is the space whose bounded functions parametrize bounded harmonic functions on .

Say has , a -dimensional isoperimetric profile, if for every finite set of vertices,

where is the set of edges joining to its complement.

For instance, Varopoulos showed that the Cayley graph of a finitely generated group does not have

3. Result

Theorem 2 Assume that has . Then, there exists a linear map

defined for , mapping constants to constants, bounded functions to bounded functions, to for all , and such that

Corollary 3 If is amenable, its Cayley graph satisfies for all .

Lemma 4 (Holopainen-Soardi)

Groups without non constant harmonic functions are called Liouville. They are all amenable. This class includes polycyclic groups, groups of intermediate growth, lamplighter over , lamplighter over with a Liouville group of lamps.

On the other hand, lamplighter over is not Liouville

Corollary 5 If is Liouville, then for all .

This fails for . Indeed, is the set of functions on the ends mod constants.

4. Proof

Let denote the probability to reach in steps from . For a function on vertices, let